With considerable zeal, Boston teacher Molly Sangalang delved into one of the more difficult math lessons for fifth-graders: multiplying decimals without a calculator.

She demonstrated examples several times on a dry-erase board, repeatedly asked students questions to gauge their understanding. Many were stumped — not the result Sangaland was looking for this particular afternoon, as her supervisor sat in the back of the Blackstone Elementary School classroom, watching her every move.

Across Massachusetts, administrators are increasingly visting classrooms this year and amassing a stockpile of notes, lesson plans, and examples of student work as they carefully judge the effectiveness of more than 68,000 teachers statewide.

The goal of the new ramped-up evaluation systems — developed under hard-fought state regulations — is to build a more skilled teaching force that can help students reach new heights. The regulations, which also apply to administrators and superintendents, encourage sharing successful teaching strategies, creating improvement plans for unsatisfactory educators, and terminating those repeatedly deemed ineffective.

“They know we are not looking for faults — we are looking for effective practices,” said Lisa Lineweaver, the Blackstone’s director of accelerated improvement and Sangalang’s supervisor. “Part of the responsibility of being in a teacher’s classroom is not making them feel nervous. . . . There are so many things a teacher is trying to do in every lesson. It’s hard work.”

Still, in Boston — where school leaders and union officials clashed for more than two years on creating a new evaluation system before reaching an agreement last fall — the change is still generating unease.

“There is a sense the district is too quick to put people on failing plans,” said Richard Stutman, referring to teachers who are rated unsatisfactory. “No one is quite sure how this will work out.”

But Superintendent Carol R. Johnson defended the evaluation system.

“We are definitely successful in implementing it so far,” Johnson said. “Our school leaders are working incredibly hard.”

The new approach is a far cry from the old way teachers had been evaluated in Massachusetts. For decades, most districts treated evaluations as a dreaded punch-list item and would observe teachers in announced visits about twice a year, deeming teachers satisfactory or unsatisfactory.

Now, administrators can conduct evaluations without prior notice, and state regulations encourage multiple visits. Educators can receive one of four ratings: exemplary, proficient, needs improvement, or unsatisfactory.

“There is great variation within the effectiveness of our teaching and adminstrative forces,” said Mitchell Chester, state commissioner of elementary and secondary education. “We have teachers who secure as much as two years of learning growth from their students, but we have other teachers who consistently get less than a year’s gain from their students.”

The Blackstone — a state-designated “underperforming” school in Boston’s South End that has been raising student achievement — emphasizes the coaching experience the new system can foster between administrators and teachers as they fine-tune classroom instruction. The school’s 52 teachers are evaluated 10 times a year by either the principal or three other administrators.

Sangalang, an accomplished teacher, said she is unfazed by the Big-Brother-like attention the evaluation system can generate, and finds the feedback useful. After the most recent session, Lineweaver dropped a note in Sangalang’s mailbox, offering feedback and requesting an update on the lesson.

“Her feedback is very targeted and specific: ‘You said this and the impact was this,’ ” she said. “I do feel there is more of a coaching aspect to the new evaluation system. . . . It’s always focused on instruction and helping me improve my craft.”

The state will up the ante of the new evaluation systems next fall when districts will be required to use student test scores to rate the performance of educators — a practice strongly pushed by President Obama.

The regulations have been a lightning rod among local unions, some of which are unhappy with leaders of the Massachusetts Teachers Association, the state’s largest teachers union, which helped craft the regulations and endorsed using standardized test scores.

About two dozen school districts have failed to reach agreements with their unions to implement the changes. Chester sent letters last month to most of those districts, which include Lowell and Lynn, informing them the delay could jeopardize their funding under the federal Race to the Top education grant program.

Boston tested out the new system last school year at 11 schools, including the Blackstone. The experiment revealed a largely well-skilled teaching force at those schools. Of the 527 teachers evaluated last year, 12 percent received an overall rating of exemplary; 77 percent were proficient; 9 percent, needs improvement; and 1 percent, unsatisfactory.

At the Blackstone, principal Cynthia Paris-Jeffries’ makes it abundantly clear she sees the evaluations as a top priority for her school of 600 children. She has “wallpapered” her office with printouts on the criteria for assessing teachers and examples of best practices evaluators should observe, such as well-structured lessons and instruction tailored to meet the individual needs of each student.

The evaluators also draw blocks next to a teacher’s name on a sheet of paper that hangs in Paris-Jeffries office for each completed observation. They place a plastic block in a big clear bowl next to the principal’s desk — giving an overall numerical tally of the observations conducted so far for all teachers. On Friday, a note on the bowl read 320.

“We are pretty aggressive about it,” Paris-Jeffries said. “Our biggest leverage is to improve teacher performance, which in turn will improve student achievement.”

One of the biggest challenges of the system, Paris-Jeffries said, is finding constructive feedback for teachers who excel at their craft.

Sangalang already knew that Lineweaver would be dropping by on a recent afternoon to observe the math lesson. Line­weaver arrived shortly after the students came back from lunch, and quietly took a seat at a table in the back of the classroom and opened up her laptop.

The lesson — multiplying decimals — was already in progress.

Lineweaver was an active observer, nodding her head several times as Sangalang explained the lesson. She also circulated among the students to quiz their comprehension.

It was during these exchanges with students that Lineweaver noticed they were struggling with the decimals. She shared her observations briefly with Sangalang, who agreed and said she was planning to work with them in small groups.

Later in her office, Line­weaver combed through more than 5½ pages of notes from the 45-minute observation, as she wrote a “rapid response memo.” She highlighted two positives: articulating a clear goal for the lesson and repeatedly checking for student understanding.

She also asked for an update on the small groups. Sangalang provided it the next day — along with a confession.

“I didn’t do the small groups because it was 2:30,” Sangalang said. “They were struggling in a good way, but we were fried. So I switched gears and came back to it today. The math coach was here, and a City Year volunteer was here too. . . . They’re definitely closer to understanding it.’’